 
Summary: SOME PROBLEMS ON FINSLER GEOMETRY
J.C. ´ALVAREZ PAIVA
Abstract. This article is an unorthodox survey of Finsler geometry
presenting both results and open problems. It aims to show that re
cent progress in convex geometry, the calculus of variations, symplectic
geometry, and integral geometry can be powerful tools in the study of
Finsler manifolds; and that Finsler geometry can prove useful in solving
some of the open problems in these fields.
We do like intuitive geometric arguments and uncovering
simple geometric reasons underlying seemingly recondite facts.
H. Busemann.
Contents
1. Introduction 2
2. Preliminaries 3
3. Volume and area in Finsler spaces 8
4. Unit spheres in Minkowski spaces 11
5. Symplectic equivalence of Finsler manifolds 14
6. Around Hilbert's fourth problem 18
7. Closed geodesics 22
8. Differential invariants of Finsler surfaces 22
